| L(s) = 1 | + 32·2-s + 640·4-s + 516·5-s + 1.37e3·7-s + 1.02e4·8-s + 1.65e4·10-s + 4.47e3·11-s + 6.69e3·13-s + 4.39e4·14-s + 1.43e5·16-s + 1.75e4·17-s + 3.14e4·19-s + 3.30e5·20-s + 1.43e5·22-s + 5.17e4·23-s + 7.98e3·25-s + 2.14e5·26-s + 8.78e5·28-s + 1.28e5·29-s + 1.50e5·31-s + 1.83e6·32-s + 5.62e5·34-s + 7.07e5·35-s + 2.09e5·37-s + 1.00e6·38-s + 5.28e6·40-s + 2.96e5·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 1.84·5-s + 1.51·7-s + 7.07·8-s + 5.22·10-s + 1.01·11-s + 0.845·13-s + 4.27·14-s + 35/4·16-s + 0.867·17-s + 1.05·19-s + 9.23·20-s + 2.86·22-s + 0.886·23-s + 0.102·25-s + 2.39·26-s + 7.55·28-s + 0.979·29-s + 0.905·31-s + 9.89·32-s + 2.45·34-s + 2.79·35-s + 0.680·37-s + 2.97·38-s + 13.0·40-s + 0.671·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(417.7392371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(417.7392371\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 516 T + 10331 p^{2} T^{2} - 3652632 p^{2} T^{3} + 251221548 p^{3} T^{4} - 3652632 p^{9} T^{5} + 10331 p^{16} T^{6} - 516 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 4470 T + 39065732 T^{2} - 104726920230 T^{3} + 870369672379398 T^{4} - 104726920230 p^{7} T^{5} + 39065732 p^{14} T^{6} - 4470 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 6698 T + 43009519 T^{2} - 593897189168 T^{3} + 8886230554631548 T^{4} - 593897189168 p^{7} T^{5} + 43009519 p^{14} T^{6} - 6698 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 17568 T + 700765970 T^{2} - 13095388763712 T^{3} + 489520096652627475 T^{4} - 13095388763712 p^{7} T^{5} + 700765970 p^{14} T^{6} - 17568 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 31490 T + 2792256832 T^{2} - 74778607610714 T^{3} + 3526368047999851246 T^{4} - 74778607610714 p^{7} T^{5} + 2792256832 p^{14} T^{6} - 31490 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 51744 T + 12458773091 T^{2} - 476623412313084 T^{3} + 62254397880132423396 T^{4} - 476623412313084 p^{7} T^{5} + 12458773091 p^{14} T^{6} - 51744 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 128652 T + 27644089661 T^{2} - 4795746792083262 T^{3} + \)\(47\!\cdots\!20\)\( T^{4} - 4795746792083262 p^{7} T^{5} + 27644089661 p^{14} T^{6} - 128652 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 150248 T + 47319056221 T^{2} + 923558254655866 T^{3} + \)\(48\!\cdots\!80\)\( T^{4} + 923558254655866 p^{7} T^{5} + 47319056221 p^{14} T^{6} - 150248 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 209726 T + 345897066025 T^{2} - 55508095827198398 T^{3} + \)\(47\!\cdots\!88\)\( T^{4} - 55508095827198398 p^{7} T^{5} + 345897066025 p^{14} T^{6} - 209726 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 296436 T + 310596069323 T^{2} - 115985004536902284 T^{3} + \)\(97\!\cdots\!12\)\( T^{4} - 115985004536902284 p^{7} T^{5} + 310596069323 p^{14} T^{6} - 296436 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 257036 T + 454904067382 T^{2} - 99781506435602432 T^{3} + \)\(18\!\cdots\!43\)\( T^{4} - 99781506435602432 p^{7} T^{5} + 454904067382 p^{14} T^{6} - 257036 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 627522 T + 1001285673245 T^{2} - 634076489905794810 T^{3} + \)\(79\!\cdots\!44\)\( T^{4} - 634076489905794810 p^{7} T^{5} + 1001285673245 p^{14} T^{6} - 627522 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 487962 T + 1151006687141 T^{2} - 1584789113685830562 T^{3} + \)\(16\!\cdots\!64\)\( T^{4} - 1584789113685830562 p^{7} T^{5} + 1151006687141 p^{14} T^{6} - 487962 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 781752 T + 7973728009202 T^{2} - 5575083945920359776 T^{3} + \)\(27\!\cdots\!99\)\( T^{4} - 5575083945920359776 p^{7} T^{5} + 7973728009202 p^{14} T^{6} - 781752 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 326894 T + 10465337869792 T^{2} - 3483646162808018330 T^{3} + \)\(46\!\cdots\!46\)\( T^{4} - 3483646162808018330 p^{7} T^{5} + 10465337869792 p^{14} T^{6} - 326894 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1481564 T + 9142557036907 T^{2} - 18680799675898954496 T^{3} + \)\(70\!\cdots\!64\)\( T^{4} - 18680799675898954496 p^{7} T^{5} + 9142557036907 p^{14} T^{6} - 1481564 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 3059352 T + 25727425581551 T^{2} - 63660948230280633588 T^{3} + \)\(30\!\cdots\!36\)\( T^{4} - 63660948230280633588 p^{7} T^{5} + 25727425581551 p^{14} T^{6} - 3059352 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1457828 T - 479590045244 T^{2} - 10379301751389344636 T^{3} + \)\(12\!\cdots\!78\)\( T^{4} - 10379301751389344636 p^{7} T^{5} - 479590045244 p^{14} T^{6} - 1457828 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 160816 T + 56649658486939 T^{2} + 55259948203637731252 T^{3} + \)\(14\!\cdots\!00\)\( T^{4} + 55259948203637731252 p^{7} T^{5} + 56649658486939 p^{14} T^{6} + 160816 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 4936182 T + 93786616943225 T^{2} - \)\(39\!\cdots\!58\)\( T^{3} + \)\(36\!\cdots\!96\)\( T^{4} - \)\(39\!\cdots\!58\)\( p^{7} T^{5} + 93786616943225 p^{14} T^{6} - 4936182 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 9236844 T + 165803811182303 T^{2} - \)\(10\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!80\)\( T^{4} - \)\(10\!\cdots\!08\)\( p^{7} T^{5} + 165803811182303 p^{14} T^{6} - 9236844 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 5284066 T + 54323204365780 T^{2} + \)\(57\!\cdots\!66\)\( T^{3} + \)\(12\!\cdots\!46\)\( T^{4} + \)\(57\!\cdots\!66\)\( p^{7} T^{5} + 54323204365780 p^{14} T^{6} + 5284066 p^{21} T^{7} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.00224146492618412155019977905, −6.30041120852375623313210817074, −6.25137497676758758619372653700, −6.18860361142421071177528142744, −6.05337059940066423513346165275, −5.63915521770264236792640196297, −5.32427033679986566679452697716, −5.22673498599694125209375166419, −5.17342122900754730104582543182, −4.70015688233957798237809924837, −4.29734593875078342955972540492, −4.24206978426140412721179238418, −4.17416032593790729056354148926, −3.42182921980683087684200256213, −3.39235704497563147253592379981, −3.22789618999478694887699704375, −2.81268406027506361587071375902, −2.34215316159125583344586243464, −2.07486154273179626329904892619, −2.04942934444790051564737708384, −1.79312308873785312206060850496, −1.29034948768753295732658846500, −0.913554575621357679910199322964, −0.836265014971809306616532991135, −0.799598116195284654487844217420,
0.799598116195284654487844217420, 0.836265014971809306616532991135, 0.913554575621357679910199322964, 1.29034948768753295732658846500, 1.79312308873785312206060850496, 2.04942934444790051564737708384, 2.07486154273179626329904892619, 2.34215316159125583344586243464, 2.81268406027506361587071375902, 3.22789618999478694887699704375, 3.39235704497563147253592379981, 3.42182921980683087684200256213, 4.17416032593790729056354148926, 4.24206978426140412721179238418, 4.29734593875078342955972540492, 4.70015688233957798237809924837, 5.17342122900754730104582543182, 5.22673498599694125209375166419, 5.32427033679986566679452697716, 5.63915521770264236792640196297, 6.05337059940066423513346165275, 6.18860361142421071177528142744, 6.25137497676758758619372653700, 6.30041120852375623313210817074, 7.00224146492618412155019977905
